Dynatomic curve and core entropy for iteration of polynomials
Dynatomic curve and core entropy for iteration of polynomials
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发表时间：
2013-04
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通讯作者：
Yan Gao
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作者：
Yan Gao
在研究由多项式族生成的动力系统时，自然会出现含有周期点或准周期点的分圆型代数曲线。本文第一章证明了在周期族fc（z）= zd + c的情形下，所有这些曲线都是光滑且不可约的，从而将已知结果推广到d = 2的情形。在同一族的预周期情形下，本文第二章证明了，与所有预期相反，这些曲线一般是可约的。此外，还讨论了不可约分量的特征及其解析几何关系。本文的第二个主题是由W. Thurston，它是多项式的核心熵。瑟斯顿给出了一个计算这些熵的算法，但没有证明。本文对该算法进行了严格的证明，并从多个角度研究了这些熵的变化规律。本文的最后一个主题给出了一类有理映射的Julia集上有C1-弧的充要条件。
When studying dynamical systems generated by a family of polynomials, it arises naturally cyclotomic type algebraic curves containing periodic or preperiodic points. In the periodic case of the family fc(z) = zd + c, the first chapter of this thesis shows that all these curves are smooth and irreducible, generalizing the known results to the case d = 2. In the preperiodic case of the same family, the second chapter of this thesis shows, against all expected that these curves are in general reducible. In addition, there contains a characterization of irreducible components and their analytical and geometrical relationship. The second theme of this thesis a new topic developed by W. Thurston, it is core entropy of polynomials. Thurston gave an algorithm, without proof, for compute these entropies. The thesis contains a rigorous proof of this algorithm and new methods to study the variation of these entropies from several views. The last topic of this thesis gives a necessary and sufficient condition for a kind of rational map having a C1-arc in its Julia set.